BIOMAT 2008

The following sections are included:

• PREFACE

• EDITORIAL BOARD OF THE BIOMAT CONSORTIUM

• CONTENTS

Systematic studies of reaction-diffusion waves begin in the 1930s with the works by Zeldovich - Frank-Kamenetskii in combustion theory and by Fisher and Kolmogorov - Petrovskii - Piskunov in population dynamics. The theory of reaction-diffusion waves is now well developed and includes detailed analysis of the scalar reaction-diffusion equations and monotone systems, of flame stability and nonlinear dynamics, of waves in excitable media, of reaction-diffusion-convection waves, and so on. After a short review of the theory of reaction-diffusion waves, some recent developments in the theory of flame propagation will be presented. They concern some mathematical aspects of combustion waves with the Lewis number different from 1 and reaction-diffusion-convection waves. Models in population dynamics with intra-specific competition will be discussed. They provide a new mechanism of pattern formation in biology and can be used to describe the emergence of biological species in the process of evolution.

Many reaction-diffusion equations exhibit front solutions. These solutions connect two asymptotic states and propagates at a constant speed. The main problem is to compute the speed of the front selected by the dynamics. The conventional methods rely on the computation of exact solutions. However, these solutions can not be computed in general. A review of the different techniques and known bounds are given and a detailed analysis of a 2D reaction-diffusion equation is given.

Biological populations can be faced with two detriments simultaneously if they experience both parasitism and an Allee effect. While infection with disease causes additional mortality, the Allee effect is a demographic process describing depensation (i.e., population decline or reduced population growth at low densities in case of a 'strong' or 'weak' Allee effect, respectively). The joint interplay of disease spread and a strong Allee effect are investigated in mathematical models that consist of two differential equations (describing the susceptible and infectious part of the host population) with a cubic nonlinearity (modelling the Allee effect). Two different incidences are considered, namely frequency-and density-dependent transmission, which model the infection process at two opposite ends of a spectrum of possibilities. Various threshold quantities are derived and employed to explain infection disappearance, parasite invasion and host extinction. The comparison of dynamical behaviour in both models provides interesting insight how depensation and disease transmission interact at various population densities. The general impact of disease is (i) to depress the host population size in endemic equilibrium and (ii) to enlarge the likelihood of extinction. If the incidence is density-dependent, oscillatory dynamics are possible as well as the emergence of three endemic equilibria, rendering the population tristable. The latter scenario is discussed in detail with respect to implications for the conservation of endangered species and the management of pests such as invasive alien species. Critical parameter values are identified for which population persistence might be possible even at extremely large values of the basic reproduction number , which could be expected to drive the host extinct independent of the initial condition.

We assumed a population affected by a disease, whose infection process is associated to a sequence of social punctual events. The event is a kind of cultural activity or an economic necessity that happens with some frequency. We formulate a generalist mathematical model for determining, with analytic techniques of the Impulsive Differential Equations, the dynamic behavior of the infectious group. We introduce diverse conditions on the frequency of the infection events with the intention of to put control tools in hands of the regulatory authority, for a better sanitary management. The idea is to avoid a spread of the disease, trying to keep the amount of infectious under predetermined levels or going towards extinction.

Understanding and being able to predict the geo-temporal pattern of epidemic dynamics would provide key information in the identification and planning of control strategies. Relying on theoretical approaches and numerical simulations, it is possible to study specific aspects of the epidemic evolution and investigate the features of human mobility that are responsible for the observed patterns. We consider a stochastic metapopulation model for the spread of an infectious disease in a spatially structured population. The model assumes strong mixing of individuals within local communities or subpopulations, and weaker interactions between people belonging to distinct subpopulations. The underlying structure of the metapopulation model connecting the subpopulations represents the patterns of human mobility and travel. Here we focus on the impact that spatial, topological and traffic properties of the metapopulation network have on the geo-temporal propagation of the disease in the system. We find that the metapopulation network topology (hop distance) is responsible for driving the invasion dynamics in waves which simultaneously affect shells of subpopulations at the same topological distance from the epidemic seed, regardless of the geographical distance. Heterogeneous travel fluxes on the other hand affect this picture by weakening the waves synchrony. Results show how the epidemic peak times are in the latter case less correlated with the topological distance, and are correlated instead with weighted paths based on the travel probabilities. Epidemic spreading in metapopulation models of spatially structured populations connected by a mobility network naturally leads to epidemic waves affecting subpopulations at the same topological distance which experience simultaneous epidemic outbreaks. A variety of factors such as multiple legs travel, heterogeneity of travel fluxes and correlations between fluxes and topology might enhance or impede the waves synchrony.

We show that the usual Kaspar-Klug classification of icosahedral viral capsids based on the triangular number T can be refined if one takes into account the differentiation of hexagonal capsomers. These can appear in three different kinds according to their symmetry axis: two-fold and chiral (abcabc), three-fold (ababab), or non-symmetric, i.e. totally differentiated (abcdef). The icosahedral capsids can be then subdivided into four symmetry classes, which in turn may be chiral or non-chiral.

We discuss the properties of the "periodic table"resulting from this classification and explore the hints concerning the mutations and the evolutionary trends of icosahedral capsid viruses.

Routine taxonomic identification is a limitation factor in the study of macroinvertebrates communities, a key group of freshwater ecosystems. Traditionally, macroinvertebrates has been identified through examination under stereoscopic microscope, an activity that requires high technical expertise and a considerable amount of time. In this paper we present the first automatic taxonomic identification of freshwater macroinvertebrate taxa, achieved through a novel image processing program developed with MATLAB^®. The program works in a completely automated fashion once it has been trained, with no user intervention. A set of morphological and texture parameters are calculated through image analysis and processed by a hierarchical set of partitioned artificial neural networks (ANNs) in order to identify the taxon to which presented specimens belong. Classification performance is estimated by 10-fold stratified cross-validation. Specimens of 10 macroinvertebrate taxa of varying taxonomic hierarchy were isolated and identified from field samples. Digital images of specimens were acquired with a flatbed scanner, yielding a database of 1042 images. Overall recognition performance was > 74% for all taxa, with most values in the 80-90% range, and a highest value of 99.48%. The followed scheme of image processing and hierarchical partitioned ANNs analysis proved to be effective for this particular challenge of pattern recognition, yielding a global classification performance of 87.83% and being able to distinguish between species of the same genus.

In this work, the HP model and Monte Carlo Method are used to study the effect hydrophobic on the folding problem. We used two lattices models (square and cubic) and several chains with distinct proportions of hydrophobic residues. We investigate how the hydrophobic residues number of the chains can influence its folding. For each simulation, we measure three parameters: Energy, End-to-End Distance and Radius of Gyration. The geometry of the final chains was analyzed too. The simulations show that the proportion of hydrophobic residues in the chain is very important for the folding. New simulations have been showed that the position of theses residues is important to the chain.

Some organisms employ multiple defence strategies against their enemies, while others fail to employ a defence that seems obvious. We shall investigate three questions for host-parasite systems.

(1) Under what circumstances does it pay for a host to employ a given defence strategy against one of its parasites?

(2) If alternative strategies are available, how is the appropriate strategy chosen?

(3) When is it appropriate to employ multiple defence strategies against an enemy?

We shall illustrate our results in two cases of brood parasites and their hosts. The paper by Britton et al. (2007) contains more background details on the basic model and the analysis but the extensions to the model and some of the results are new.

The notion that words compete and languages evolve in analogy to individuals and populations was already familiar in the nineteenth century as expressed in this quotation by the famous Darwin contemporary philologist Max Müller, "A struggle for life is constantly going on amongst the words and grammatical forms in each language. The better, the shorter, the easier forms are constantly gaining the upper hand, and they owe their success to their own inherent virtue." A more suitable analogue to language, however, is that of a parasitic species since language does not exist without speakers, just like parasites do not exist without hosts. Indeed, the view of language as a purely cultural trait which follows the rules of cultural rather than of biological evolution leads to a mathematical description of language evolution very similar to the formulation of the population dynamics of infectious agents, since the transmission of the language occurs only through the direct interaction between language-proficient (i.e., infected) adults and children. Here we use the results of recent experiments on infants, which demonstrate that they are imprinted by the language of their parents so as to favor contact with individuals that speak the same language, to replace the usual assumption of random meeting between individuals by a procedure in which children born from language-proficient parents but who have failed to learn the language from them (vertical transmission), can actively search for unrelated adults (oblique transmission) that speak the same language. We find that by properly setting the parameters that control the efficiency of oblique transmission, language can be maintained in the population even if it lacks of adaptive value.

Among the many factors that contribute to overexploitation of marine fisheries, the role played by uncertainty is important. This uncertainty includes both the scientific uncertainties related to the resource dynamics and the evolution of its price. Recently, many works advocate for the use of Marine Protected Areas (MPAs) as a central element of future stock management. In this work we investigate and analyse the impacts of the creation of MPAs, in economic sustainability through a bioeconomical model integrating the evolution of the resource price. Equilibria and stability of the model are studied. Also, instead of studying the environmental and economic interactions in terms of optimal control, we focus on the viability of the system. This viability is defined by a set of economic state constraints. This constraints combine a guaranteed consumption and a minimum income for fishermen. Using the mathematical concept of viability kernel, we exhibit how marines reserves might guarantee a perennial system and viable fisheries.

We present a mathematical model which couples a population growth dynamic subject to an Allee effect and a long distance dispersal process. First we analyzed the local dynamic through the equilibria and their stability. For the spatial dynamic we used numerical simulations, that allowed us to observe the spatial expansion of the population and to track the spatial displacement of the invasion front. This permitted us to calculated the expansion speeds. We determined the influence of the Allee effect, reproductive capacity and the long distance dispersal on the invasion speeds. We observed that an Allee effect turns accelerating expansion speeds into constant speeds. Expansion speeds decreases with Allee effect intensity but increases with the reproductive capacity of the population. Our results show that While dispersal contributes to expansion speeds, it also turns the population more susceptible to extinction.

Malaria has emerged as a frequent problem in international travelers. The risk depends on destination, duration and season of travel. However, data to quantify the true risk for travelers to acquire malaria are lacking. Methods: We used mathematical models to estimate the risk of non-immune persons to acquire malaria when traveling to the Amazon region. From the force of infection we calculated the risk of dengue dependent on duration of stay and season of arrival. Our data highlight that the risk for non-immune travelers to acquire malaria in the Amazon region is substantial but varies greatly with seasons and epidemic cycles. For instance, for a traveler who stays in the Amazon for 120 days during the high season, the risk of acquiring malaria was 0.16%. Risk estimates based on mathematical modelling will help the travel medicine provider give better evidence based advice for travelers to malarial countries.

Neural Networks are efficient classification tools that have been applied to several applications including extracting regularities in data and classifying events in finance, marketing, internet and biomedicine. The training process uses available examples to produce a model and classify new events based on the extracted model. This learning procedure based on the examples is not capable of taking prior knowledge that is either available or discovered in data into account. In the present work, we propose a way to include prior knowledge into Radial Basis Function Neural Networks and to express the knowledge as a set of linear constraints in the least square problem. The obtained method still takes advantage of kernel functions to obtain a nonlinear classifier. Furthermore, its computational complexity is not affected while the misclassification error is enhanced. Publicly available biomedical datasets are used in a case study to analyze the performance of the approach, and to compare the results with the state of the art classifiers.

Despite of great interest and numerous analysis assays the knowledge of pure mutation process is still scarce and insufficient. The aim of our investigation was to connect environmental conditions and some specific trends in substitutions patterns observed in different bacterial genomes. As a tool for genome large scale analysis, based on Borrelia burgdorferi B31 chromosomal and plasmid complete sequences and 13 others bacterial chromosomes complete sequences available at the NCBI FTP site, we used Markov chains. We assumed that pure mutational pressure could be considered as Markovian process hence substitution patterns could be described as a Markov chain transition matrix. We attempt to answer the question if its ergodic state reflects the nucleotide composition of the given sequences equilibrium state and if it could characterize the direction of mutational pressure specific for a particular genome.

The phylogenetic inference problem consist of determining the best evolutionary relation among species.³¹ There are several methods proposed in the literature which resolve the this problem using a optimality criterion which evaluates each possible solution.¹¹ Nevertheless, different criteria may lead to distinct phylogenies, which often conflict with each other. Moreover, other factors of phylogenetic inference may generate conflicting solutions.²⁶ In this context, a multi-objective formulation can help to overcome these incongruities. In this paper, we make a review of the main multi-objective approaches for phylogenetic inference proposed in the literature.^1,4,25

Classifiers built through supervised learning techniques are widely used in computational biology. Examples are neural networks, decision trees and support vector machines. Recently, an extension of Regularized Generalized Eigenvalues Classifier (ReGEC) has been proposed, in which prior knowledge is included. When knowledge is formalized as a set of linear constraints to the ReGEC, the resulting non linear classifier has a lower complexity and halves the misclassi-fication error with respect to the original method. In this work, we show how logic programming can extract knowledge from data to enhance classification models produced by ReGEC. The knowledge extraction method is based on two phases: a feature selection phase and a rules extraction phase. Feature selection is formulated as an integer programming problem that extends a set covering problem. The extraction phase is performed through the iterative solution of different instances of the same minimum cost satisfiability problem that models the logic separation rules used for classification. The overall method, that we call LF-ReGEC, guarantees that the number of points in the training set is not increased and the resulting model does not overfit the problem. Furthermore, the overall accuracy of the method is increased. Finally, the method is compared with other methods using genomic and proteomic data sets taken from the literature.

Recent developments in medical image analysis, phylogenetics and proteomics motivate the statistical analysis of populations of tree-structured data objects. In this context, unsupervised classification of trees arises as a challenging new area that depends on the careful development of novel mathematical framework. The discussion will center on statistical aspects of clustering in a framework where the tree data to be clustered has been sampled from some unknown probability distribution. Following Ref. 12, we will try to verify two conditions: appropriateness, the clustering of the data set should reveal some structure of the underlying data rather than model artifacts due to the random sampling process; and steadiness, the more sample points we have, the more reliable the clustering should be. We will argue about steadiness and reliability by showing an extension of the convergence properties for a class of non-parametric clustering algorithm: k-means, defined on different metric spaces of trees. We will explore the appropriateness of the clustering outputs of k-means on a real data set from proteomics, and we will comment the results from Ref. 1 on three real data sets of phylogenetic trees.

Bioinformatics has developed primarily as a discipline within mathematics and computer science devoted to organizing and analyzing large biological databases. However, biology has much to offer to a synthetic discipline of bioinformatics that draws upon and respects the mutual contributions of biology, mathematics and computer science. In particular, biology has two major theoretical foundations, both evolutionary: namely, phylogenetic systematics and population genetics, that can serve as a cornerstone of a theoretical foundation of bioinformatics along with traditional empirically driven, pattern searching forms of classical bioinformatics. In this re-conception of bioinformatics, mathematics and computer science are instrumental in developing biological theory and in solving practical biological problems. Since the genetic code is both an evolutionary product as well as a process for mediating the conversion of genotype to phenotype, it is argued here that an evolutionary analysis of genetic codes will fundamentally affect our ability to make meaning out of molecular messages through a theoretically grounded bioinformatics. How do the mathematical properties of genetic codes relate to selection pressures on the rate of synthesis of proteins, correctability and detectability of mutations, compactness of genes, and the origins of genetic codes by employing coding theory (Baudot codes, Gray codes, Hamming codes, Huffman codes, common free codes, etc.), abstract algebra, graph theory, combinatorics, information theory, efficiencies, symmetries, and phylogenetic systematics of sequences? Genetic codes become much more understandable and elegant to biologists, mathematicians, and computer scientists when they are not considered as mere ciphers, but are instead understood from three perspectives: codes as codes per se, physical chemical interactions, and evolutionary selective pressures. These various faces of genetic codes are useful for making meaning out of molecular messages, applying causal mechanisms to complex patterns, and the efficient storage and retrieval of large complex data sets. In addition, some of the alternative distance metrics based upon different mathematical representations of genetic codes that have utility in genomic data base searching (comparative sequence analyses and gene finding), phylogenetic tree construction, and prediction of three dimensional structure from primary structure will be illustrated and different evolutionary mechanisms affecting gene expression based upon codon usage will be considered.

Integrative Biology is the study of an organism within a framework in an integrated, systematic manner in order to discern governing principles or mechanisms. Quantitative tools applied in the study of biological organisms include, in addition to statistical analyzes and hypothesis testing, mathematical modeling. Computational tools used include databases to organize both the data and models into a form that is linked and readily usable. I will describe mathematical models integrated into research in physiology as well as tools being developed by the Physiome Project with the support of the International Union of Physiological Sciences. The goal of the Physiome Project is the quantitative description of the integrated function of living organisms, and for the human physiome, to develop quantitative biology to improve medical science from genes to health. The "model validate" cycle used in Mathematical Biology is iterated to refine our understanding of the biology as illustrated here with experiments, databases and modeling in kidney physiology.

In this article we studied in machina an approach to simulate the process of antigenic mutation. Our results have suggested that the durability of the immune memory is affected by the process of antigenic mutation and by populations of soluble antibodies in the blood. The results also suggest that the decrease of the production of antibodies favors the global maintenance of immune memory.

In this work, we did the construction of a fuzzy mathematical model wich it was developed to predict the pathological stage of prostate cancer.⁵ The intention is to help the specialists on the decision process about stage of the disease, to avoid surgery and intensive treatments unnecessary. The model consists on a system founded in fuzzy rules, that it combine the pre-surgical data (clinic state, PSA level and Gleason score) availing of a set of linguistic rules made with base on informations of the existents nomograms. Herewith we hoped to get the chance of the individual, with certain clinical features, be in each stage of the tumor extension: localized, advanced locally and metastatic. Simulations were made with patient's data of the Clinics Hospital/UNICAMP and the results were compared with Kattan's probabilities⁸ that are used on the medicals decisions. A software was developed from this model and is a graphic interface that makes interaction with the subroutines that make the calculations. Its source code was written in Java and software has been tested on Windows and Linux / GNU.

In opposite to the advances in computer aided surgical procedures, computer simulation of the human eye has not experienced the same technological growth. In fact, even basic questions such as the computational modeling of the structures comprising the human eye system and how to simulate the projection of an image onto the retina for a specific eye have not been addressed properly. In this work we present a framework for modeling, simulation and visualization of the human eye system. The proposed technique makes use of schematic data, and may represent a more reliable method for predicting retinal image formation given that the corneal anomalies should have greater effect in image distortion compared to the contributions of the lens and other internal media. Qualitative and quantitative validations of our approach is also presented, showing that such a methodology can be used as a virtual environment for teachers and students of ophthalmology and optometry, as well as for computer simulation of retinal images.

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